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Int. J. Electrochem. Sci., 7 (2012) 5204 - 5216
International Journal of
ELECTROCHEMICAL SCIENCE
www.electrochemsci.org
The Relationship of Surface Roughness and Work Function of
Pure Silver by Numerical Modeling
Ye Wan1,2*
, Yanbo Li1, Qing Wang
1, Ke Zhang
3, Yuhou Wu
3
1 School of Materials Science and Engineering, Shenyang Jianzhu University, Shenyang 110168, China
2 Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904,
USA 3 School of Traffic and Mechanical Engineering, Shenyang Jianzhu University, Shenyang 110168,
China *E-mail: ywan@sjzu.edu.cn
Received: 28 March 2012 / Accepted: 6 May 2012 / Published: 1 June 2012
Surface of silver has important effects on its performance in natural environments. Surface roughness
and work function of the silver samples were measured using confocal laser scanning microscopy and
scanning Kelvin probe respectively. A model concerning the relationship of the surface roughness and
work function was proposed. The objective is to provide a foundation for the effective use of
numerical model about the relationship of the surface roughness and work function. The result showed
work function decreased with surface roughness. Comparison between the experiments and the model
about roughness and work function showed the model was in agreement with the experiments. The
theoretical results of numerical model were obtained by computational approaches and based on
Maple® codes.
Keywords: Silver; Work function; Surface roughness; Modeling study
1. INTRODUCTION
Work function (or surface potential for nonmetal) is one of the important properties of solid
surfaces. Valence electrons are confined to the surface of metals by a surface potential barrier. Work
function (WF), denoted by φ, is the difference of the rise in mean electrostatic potential across the
metal surface and the bulk chemical potential of the electrons relative to the mean electrostatic
potential in the metal interior. In other words, work function is equal to the minimum work that must
be done to remove an electron from the surface of metal at 0 K [1], so it is a main factor decisive to the
corrosion properties of materials. Lower WF can improve electron activities and raise corrosion rate,
while higher WF corresponds to higher resistance to corrosion. WF can be easily measured by Kelvin
Int. J. Electrochem. Sci., Vol. 7, 2012
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probe technique, which is a noncontact, nondestructive vibrating capacitor technique used to measure
the work function difference (∆φ) between a conducting tip and a conducting sample [2].
Surface roughness (SR), one basic parameter of surface condition, influences significantly the
measured chemical, physical, dielectric, and mechanical properties of "real" materials [3-6]. Work
function, a fundamental electronic property of a metallic surface, is extremely sensitive to surface
conditions [7]. However, the study of the effect of SR on work function is rather rare. Some existing
results showed [6, 8-10] that the larger SR is, the smaller WF is. In generally, this phenomenon was
explained by SR effect on the dipole barrier [11]. At the same time, the effect of SR is often
accompanied with other factors, such as oxidation and surface treatment. After the surface was treated,
the effect of SR of the materials on their work function even showed opposite phenomena [12].
Silver is a very important metal because it has the highest electrical, the highest thermal
conductivity and the highest reflectance in the visible and infrared spectral regions of all the metals
[13]. WF of silver surface and its corrosion possibility are very important to its later usage in the
atmospheres. WF of silver surfaces and the surface potential have been reported by the photoelectric
methods [7, 14-16]. However, none of these WF investigations was an independent research made of
SR because the scientists couldn't resolve the technical problems at that time. Although there were
many phenomena showed there was some dependence of WF on silver surface, there were few
quantitative data available for SR of silver.
So far, the detailed relationship of SR and WF of silver is not clear, and a comprehensive
understanding of WF on silver is still not sufficient. In the present study, the silver surfaces with
different SR were prepared. The relationship of SR and WF of silver were investigated with a
numerical model as well. Moreover, a numerical model on the relationship between SR and WF of
silver surface was deduced on the basis of a theoretical simulation.
2. EXPERIMENTAL
2.1 Confocal laser scanning microscopy measure
Confocal laser scanning microscopy (CLSM) was provided by Carl Zeiss Inc. (Thornwood,
New York, USA) is a contact-free method [17-18] and can provide 3-D images and surface roughness.
CLSM measurement was performed at the wavelength of 543 nm. The scan area, there was 512 points
collected per line, of the instrument was set to 100 μm × 100 μm. SR can be characterized
approximately by a mean height of irregularities about an average plane, and a correlation lengthening
between irregularities [19]. The calculating method of SR for CLSM is the mean height of all surface
height values and can be given basing on the Eq. (1) [20], which is a collection of statistical and
mathematical methods that are useful to model and analyze SR.
(1)
Int. J. Electrochem. Sci., Vol. 7, 2012
5206
where Nx and Ny are the numbers of pixels in x-direction and y-direction, and is the
elevation for a given point.
2.2 Scanning Kelvin probe measure
WFs of silver samples with different surface roughness were carried out using a scanning
Kelvin probe (SKP) system provided by KP Technology Ltd. (Caithness, UK). The samples were
mounted in the SKP apparatus and allowed to equilibrate for a period of half an hour with an
atmosphere at ca. 60% relative humidity (RH). A gold plated tip with a diameter of 2 mm was used
and scanned a projected area uniformly. The tip moved 0.635 μm per step. The measurements (on
different samples) were performed with the same gradient and with the sample space within 1 micron.
The scanning area, covering 10 point ×10 point in the experiment, was 6.35 mm× 6.35 mm.
2.3 Materials
Fine silver samples of 75mm ×15mm ×0.25mm and 99.95% purity were obtained from Lucas-
Milhaupt. Inc. (Cudahy, WI, USA). Some surfaces of silver samples were wet abraded with 120, 180,
240, 400, 600, 800 and 1200 grit grinding paper respectively. The others were prepared using 1200 grit
grinding paper before they were polished with aqueous grinding paste solutions of 1 μm or 0.05 μm
alumina respectively. All the silver samples with various surface roughnesses were followed by
degreased with Alconox detergent and cleaned in deionized water, and dried in air for further
measurement of SR and WF. There were some grooves along the polishing directions (Fig. 2).
3. MODELING
Simulations were run with the Maple®
software, which solved problems with integral equations
and plotting. WFs were measured with SKP, which principle [21] was to consider the electronic energy
diagram for the surfaces of two dissimilar metals, which were separated by a distance d (shown in
Fig.1). Before electrical contact was made, such as SW was open, the highest energy electrons in each
metal occupied states at their respective Fermi level εtip and εs. and were the WFs of the tip and
the silver sample respectively, just as seen in Fig.1(a). If an external electrical contact was made
between the tip and the silver sample, seen in Fig.1(b), electrons would flow from the metal with a
lower WF to that with a higher WF until εtip equalized εs. Then Fermi levels of the two surfaces
became equal and the two surfaces were oppositely charged. The corresponding flows of electrons
produced a potential gradient (termed contact potential VC).
Given,
, (2)
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5207
where e was the electronic charge. Then,
(3)
And Q, the surface charge, was expressed as:
, (4)
The tip vibrated periodically in the space, and its vibrating function could be expressed as
. Where and were the frequency and the amplitude of oscillation of the tip
respectively. t was oscillation time. In generally, oscillation frequency of the tip of SKP was between
10 KHz - 20 KHz, which was much bigger than the moving velocity of the tip along x-direction and y-
direction.
A B
Figure 1. Electronic energy level diagrams for the tip and the sample separated by a distance d (a)
before electrical contact (SW open), and (b) after contact. εtip, εs, and , refer to the
respective Fermi levels and work function for the tip and the silver sample.
In order to make the model simple, assuming:
(a) The tip moves at a stable flat plate. That means the amplitude and the vibrating frequency of
the tip were constant. The micro capacitor formed between the length of the tip and the corresponding
area on the surface of the sample is a flat parallel-plate capacitor. It gives:
, (5)
where Ck is the capacitance of the capacitor. ε0 is the vacuum permittivity in free space (ε0
=8.85×10-12
m-3
·kg-1
·s4·A
2). εr is the relative permittivity in air (εr=1). A and d is the area and the
distance between the tip and a given surface of the silver sample respectively (shown in Fig.2).
Int. J. Electrochem. Sci., Vol. 7, 2012
5208
A B
Figure 2. (a) microscope of the silver surface abraded with 600 grit paper, and (b) cross-section
microscope of the silver surface abraded with 600 grit paper. The cross section was abraded
with successively abrasive paper before the sample was polished with 0.05 μm alumina paste.
Figure 3. Schematic view of a parallel plate capacitor between the tip and the surface of the silver
sample. Assuming the surface of silver is a periodical wave function along x-direction.
Int. J. Electrochem. Sci., Vol. 7, 2012
5209
(b) z(x, y), the elevation of a given point in the surface of the silver samples, shows a periodic
fluctuation of the x-direction distance microscopically and y-direction is parallel to the groove
direction. z(x, y) can be expressed as . d0 is the periodic peak amplitude of the
grooves created by polishing, on the silver surface (shown in Fig. 3).
It was reasonable that z was dependent of y because the abrading grooves on the surface were
parallel to each other along the abrading direction of the sample (shown in Fig. 3 and Fig. 4).
Int. J. Electrochem. Sci., Vol. 7, 2012
5210
Figure. 4 The surface images and the measured surface roughness of the samples by the CLSM
instrument. (a), (b), (c), (d), (e), (f) and (g) were those abraded with 120, 180, 240, 400, 600,
800, 1200 grit grinding paper respectively. While (h) and (i) were polished with 1μm and
0.05μm alumina grinding paste respectively.
As the area was continuous, the Eq. (1) could be expressed as the following integral function
when the laser of the CLSM instrument scanned along x- and y- directions:
(6)
S0 was the projected area on the surface of the sample. SR can be calculated from a given
scanning area of CLSM. x and y were the scanning distances of the laser in the CLSM instrument
along x-direction and y-direction respectively. The unit of Ra was determined by the CLSM instrument
and it was micron in the experiment.
Introduction of assumption (a) and (b) gave the expression (5) for a flat parallel-plate capacitor
between a small area dS0 and the opposite surface area on the tip:
, (7)
Int. J. Electrochem. Sci., Vol. 7, 2012
5211
Basing on the assumption (b),
(8)
(9)
Here, was the average distance of the flat parallel-plate capacitor between the surface of
silver sample and the gold tip (Fig. 2). and were the distances of a step of the tip of SKP along x-
direction and y-direction respectively. n0 was the steps of the tip of SKP along x-direction.
In an ideal condition, the whole surface of silver sample toward the gold tip was flat, then d0=0,
and h0=a. a was the distance of the tip to the peak of the surface of the silver samples in Fig. 3. There
is an ideal flat parallel-plate capacitor between silver sample and gold tip. The charge of the whole
surface of the tip, can be expressed as the following:
(10)
where Ck0 and A0 are the capacitance and the surface area of the ideal flat parallel-plate
capacitor between the surface of silver sample and the gold tip respectively. A0 equals the surface area
of the gold tip. and are the WFs of the tip and the silver sample below the vacuum level. Q,
the charge of the given area (S0, the area that the tip scans one step along x-direction and y-direction)
on the surface of the silver sample, is proportional to the given area:
(11)
In order to get WFs in the given area of the silver sample, we inserted Eqs. (4), (9) and (11)
into Eq.(3), giving:
(12)
(c) Because gold is a noble metal and , then,
and
(13)
Int. J. Electrochem. Sci., Vol. 7, 2012
5212
and are common for the materials of tip and samples. After is set, the expression of
and will be determined.
4. RESULTS AND DISCUSSION
4.1 Surface micromorphology
After prepared with different methods, the surfaces of the silver samples showed distinct
patterns. The surface of the silver sample abraded with finer particle size of abrasive materials was
much more burnishing than those with coarser abrasive materials. Fig.4 depicted the micro
morphology of the silver samples abraded with different particle sizes of abrasive paper or paste. It
showed that the surface of the silver sample abraded with finer particle size of grinding materials was
much smoother. There were a lot of parallel grooves along the abrading direction in the micro-
morphology, which verified the validation of the assumption (b) in section 3 as well. Surface
roughness (PSa in Fig. 4) was calculated by CLSM software after the CLSM system measured the
micromorphology. The different grit abrasive papers or pastes were key factors influencing the surface
condition of the samples. For example, Sa of the surface of the samples prepared with 120 grit grinding
paper was 3.205 μm and it was 1.289 μm for the sample with 0.05μm grinding paste. The particle sizes
of the grinding papers and pastes were in the order: 120 grit > 180 grit > 240 grit > 400 grit > 600 grit
> 800 grit > 1200 grit > 1μm > 0.05μm. The values of SR were in the same order as the particle sizes of
the grinding papers and pastes. The bigger was the particle size of the grinding paper or paste, the
bigger was the values of Sa. Sa was the biggest for the surface of the samples abraded with 120 grit
paper.
4.2 Work function of the silver samples
Table 1. The SRs and the mean WFs of silver abraded by different abrading materials
Abraded
condition
experimental SR,
μm
WF from the models ,
mV
experimental mean WF,
mV
120 grit paper 3.205 2135.2 2514
180 grit paper 2.920 2310.1 2603
240 grit paper 2.738 2422.3 2721
400 grit paper 2.657 2472.4 2825
600 grit paper 2.616 2497.8 2875
800 grit paper 2.464 2592.1 2960
1200 grit paper 2.324 2679.3 3014
1μm Al2O3 paste 1.384 3276.5 3163
0.05μm Al2O3
paste
1.289 3338.4 3176
Int. J. Electrochem. Sci., Vol. 7, 2012
5213
Table 1 showed the SRs and the WFs of the surface of the silver samples abraded with different
particle sizes of grinding paper or paste by the SKP system. It could be seen from Table 1 that the SRs
increased and the WFs decreased with the particle sizes of the abrasive materials. The values of WF
demonstrated that WF was dependent on the surface roughness of silver samples. That was because a
rougher surface had lower constraint for electrons to escape from peaks, resulting in lower WF and
consequently higher corrosion rates [22]. The results on the corrosion rate of different WF and
different surface roughness will be shown in a later research paper.
4.3 Model validation
Recall that the SR model and the calculated Sa in Eq. (6), it could be seen that there were two
main factors affecting the value of Sa. One was d0, which depended on the abrasive paper. It could be
seen from Eq. (6) that the bigger was d0, the bigger was Sa. Therefore, the bigger was the particle size
of the abrasive paper or paste, the bigger was the values of Sa, which was in agreement with the
experimental results in Fig.3 from CLSM. The other main factor affecting Sa was the scanning distance
(x) of the laser in the image instrument. The absolute value of sin(x) in Eq. (6) depended on the value
of x. SR is a relative quantity and depends on the ratio of length scale of the irregularities to the
wavelength of the laser [23], which relies on the accuracy of the instrument. Since x was set to 100 μm
in the experiment,
. (14)
However, if x is set to 635 μm,
. (15)
Eq. (9) showed the relationship of the scanning step of SKP tip and capacitance. x0, the distance
of a step of the SKP tip, was set to 0.635 μm in the CLSM instrument. Assume n0=1, then Eq. (9) can
be described as the following:
(16)
Because the tracking system of SKP could hold automatically the distance to be 0.3175μm
between the tip and the sample, and the diameter of the tip (2mm) was much bigger than the scanning
distance of the CLSM, was 0.3175μm and was . From Eq. (14), . Note
that the average surface potentials of silver and gold below the vacuum level were
and respectively [24], the model concerning the relationship between and
was as the following:
Int. J. Electrochem. Sci., Vol. 7, 2012
5214
(17)
Figure 5. Comparison of the experimental WF profiles obtained by SKP (black stars) with the
simulated WFs (blue plots, the black line is leastsquares linear fitting by those blue plots) and
the simulated curve (red line) profiles obtained by simulating numerical model at different SRs.
The green line was leastsquares linear fitting of the experimental WF of the samples abrading
from 120 grit abrading paper to 0.05μm polishing paste, while the blue line was that from 120
grit to 1200 grit abrading paper.
Fig.5 showed the simulated relationship of WF and SR of the silver surfaces abraded with
different grit papers. The simulation was run with the Maple® software. The experimental WFs were
also plotted in Fig.5. The red line was the curve drawn according to Eq. (17). The black line was the
leastsquares linear fitting curve of the blue plots calculated by Eq. (17) when SRs were from 1.289 μm
to 3.205μm. The black stars were the experimental WFs of the surfaces with different SRs and shown
in Fig.5. The green line was leastsquares linear fitting curve basing on the experimental WFs of the
surfaces with different SRs. The blue line was leastsquares linear fitting curve basing on the
experimental WFs of the surfaces with SRs from 3.205 μm to1.289 μm.
The equation for the leastsquares linear fitting was:
. (18)
Int. J. Electrochem. Sci., Vol. 7, 2012
5215
was the extreme value of WF of silver at different environment. stood for the speed of the
changes of WFs of the silver surfaces at different surface roughness. The values of , and were
listed in Table 2. was the correlation coefficient of the leastsquares linear fitting curves. The values
of showed that WF decreased linearly with in the experiments. It could be seen from Table 2 that
the changing trend of WF in Eq. (17) was very familiar with that of the experimental WF of silver
surfaces when SRs were from 3.205 μm to 2.324 μm.
Table 2. Parameters of , and of linear fitting formula
m n R
Eq (17) 4219.4 657.9 0.9999
Experimental WFs for the SRs from 2.324 to
3.205
4460.4 618.8 0.9806
Experimental WFs for all the SRs 3664.2 330 0.9302
The standard deviation (SD) of from Eq. (17) to the experimental WFs (when SRs were form
2.324 μm to 3.205 μm) was 6.3% and the model was valid at the condition. One of the reasons why
there was a deviation might be because there was periodic fluctuation of the surfaces being simulated
the charge in the capacitor. However, n from Eq. (17) deviated far away from the experimental WFs of
silver surfaces with all of the SRs. The error of was . When SR was
too big, was too big to make a parallel plates, and the plates couldn't be considered as a parallel-
plate capacitor any more. It could be seen from the values of , there were also some effects of the
boundary condition on WF of silver. The value of in Eq. (18) was closer to WFs of the surfaces
when SRs were from 1.289 μm to 3.205 μm than that with SRs from 2.324 μm to 3.205μm. The
smoother was the surface and the smaller was the difference of between the model and the
experimental values. If the grinding paper were too coarse or too fine, there were the boundary
conditions to calculate WF in the model.
WF could be interpreted easily from surface roughness of silver. If the surface of silver was
rough, the electrons on the surface of the silver samples were easy to depart from the peaks of the
grooves. That is, it kept a lower WF. Since the grooves were parallel to the grinding direction (shown
in Fig. 2 and Fig. 4) of the samples, a uniform distribution of WF along the grinding direction was
expected. WFs measured in SKP system also verified the results and the validation of the assumption
(b) in the section 3. It was noted that the simulated values were lower than the experimental ones. The
difference between simulated values and experimental values might be attributed to the surrounding
environments. The WFs from Eq. (16) were based on the average surface potentials of silver and gold
below the vacuum level. However, the experimental WFs were in the 60% RH atmosphere. These
should be another reason there were some difference between WF from the model and the
experiments.
Int. J. Electrochem. Sci., Vol. 7, 2012
5216
5. CONCLUSIONS
Based on the theories of flat parallel-plate capacitor and the surface roughness, a numerical
model of the relationship of WF and SR was deduced. The numerical model revealed that WFs of the
surfaces of the silver samples decreased linearly with Sa. The experimental results demonstrated that
WF decreased linearly with SR of silver surface. The numerical model deduced from the software
might be used to detect the value of SR or WF after one of them was decided, and could be used to
calculate the depth of the groove on the polished surface of the silver samples as well.
ACKNOWLEDGMENTS
This work was supported by grants from D. Dunmire, Director of Corrosion Policy and Oversight,
Office of the Secretary of Defense, NSFC Projects 51101106 and 51131007. The discussions with
Prof. Kelly are gratefully acknowledged. Special thanks go to Dr. George W.G. Booth about WF,
Wasiu Adedeji about the discussion of surface roughness and Lok-kun Tsui for the usage of Maple
software and the debugging of the model.
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